**Towering
Mathematics**

In a recent visit to a seventh-grade
mathematics classroom, I was delighted to see students working collaboratively
on the Skeleton Tower task as shown in **figure
1**.

**Fig. 1** How many cubes
are needed to build a tower like this but 12 cubes high? (*Source:* http://www.illustrativemathematics.org/illustrations/75)

The task is well suited
for exploration with middle-grades students. The problem lends itself to
multiple solution strategies and multiple representations and can be explored
with a variety of tools, including manipulative cubes, spreadsheets, and design
software (e.g., see NCTM’s Isometric Drawing Tool at http://illuminations.nctm.org/Activity.aspx?id=4182).

**Student Approaches**

In small groups, many seventh
graders broke the tower into 5 pieces. They saw a center column 12 cubes tall
and 4 identical “legs,” each consisting of 1 + 2 + 3 + … + 9 + 10 + 11 = 66
cubes. Ultimately, these students calculated that a Skeleton Tower with height
12 units would require 12 + 4 ´ 66 = 276 cubes. Others saw different
patterns. For instance, one seventh grader told me that the problem essentially
involved “finding the area of a rectangle.” Creatively rearranging cubes, she
noted that “12 times 25” cubes were needed for the tower of height 12. Following
up, I asked this student how many cubes would be needed for a tower 100 cubes
tall. She quickly replied, “Easy! 100 ´ 199.” Then she
showed a diagram similar to the one below.

Do
you see what she saw?

The Skeleton Tower task
is well aligned with the Standards for Mathematical Practice, in particular, 7.
Look for and make use of structure. Note that both solution strategies
described here highlight students’ use of structure, both numerical and
geometric, to solve the task. In the first approach, students sum consecutive
integers to find a solution, which is an approach intimately linked to triangular numbers and Gauss.
For instance, students can calculate 1 + 2 + 3 + … + 9 + 10 + 11 by pairing
addends creatively.

1 + 2 + 3 + 4 + 5 + 6 +
7 + 8 + 9 + 10 + 11 = (0 + 11) + (1 + 10) + (2 + 9) + (3 + 8) + (4 + 7) + (5 + 6)
= 6 ´
11 = 66

In the second approach,
students rearrange blocks to create an easier shape, a 12 ´
(11 + 11 + 1) rectangle, to analyze. Both approaches illustrate the suitability
of the task for a wide range of learners. Both strategies hint at
generalizations that may be explored with more advanced students. Meanwhile,
the task’s concrete context makes it a meaningful problem for students to
explore with hands-on materials.

Michael
Todd Edwards is an associate professor of mathematics education at Miami
University in Oxford, Ohio. He is the coeditor of *Contemporary Issues in Technology and Mathematics Teacher Education*,
executive editor of the *North American
GeoGebra Journal*, and codirector of the GeoGebra Institute of Ohio. His
research interests focus on the teaching and learning of school mathematics
with technology (specifically, dynamic mathematics software), ethical issues
surrounding the use of free software and the free software movement, and
writing as a vehicle to learn mathematics at all levels of instruction.